Problems and Results

The notion of weak visibility of a polygon from a segment was introduced by Avis and Toussaint [42] in the context of the art gallery problem. They considered a variation of the problem when there is only one guard and the guard is permitted to move along an edge of the polygon. They defined visibility from an edge vivi+1 in a simple polygon P in three different ways.

1. P is said to be completely visible from viυi+1 if every point z ∈ P and any point w ∈ vivi+1, w and z are visible (Figure 3.1(a)).

2. P is said to be strongly visible from vivi+1 if there exists a point w ∈ vivi+1 such that for every point z ∈ P, w and z are visible (Figure 3.1(b)).

3. P is said to be weakly visible from vivi+1 if each point z ∈ P, there exists a point w ∈ vivi+1 (depending on z) such that w and z are visible (Figure 3.1(c)).

If P is completely visible from vivi+1, the guard can be positioned at any point w on vivi+1. In other words, P and the visibility polygon V(w) of P from any point w ∈ vivi+1 are same. If P is strongly visible from vivi+1, there exists at least one point w on vivi+1 from which the guard can see the entire P, i.e., P = V(w). Finally, if P is weakly visible from vivi+1, it is necessary for the guard to patrol along vivi+1 in order to see the entire P.